Borel equivalence relations induced by actions of tsi Polish groups

Abstract: We study Borel equivalence relations induced by Borel actions of tsi Polish groups on standard Borel spaces. We characterize when such an equivalence relation admits classification by countable structures using a variant of the $\mathbb G_0$-dichotomy. In particular, we find a class that serves as a base for non-classification by countable structures for these equivalence relations under Borel reducibility. We use this characterization together with the result of [B. D. Miller, to appear in the Journal of Mathematical Logic] to show that if such an equivalence relation admits classification by countable structures but it is not essentially countable, then the equivalence relation ${\mathbb E}^{\mathbb N}_0=\mathbb E_3$ Borel reduces to it.